RUI: Hurwitz Number Fields

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Two of the oldest areas of mathematics are number theory and geometry. Number fields play a central role in number theory, while surfaces play a central role in geometry. The research project concerns number fields that arise naturally from the geometry of surfaces, via a construction dating back to work of the mathematician Adolph Hurwitz around 1900. These Hurwitz number fields have a combination of properties that is very unusual from a purely number-theoretic point of view: their degrees and Galois groups are typically very large, while their sets of ramifying primes are very small. In fact, Hurwitz number fields are so extreme with respect to these invariants that their existence contradicts a heuristic that has successfully guided other recent research into number fields. The goal of this research project is to establish that these outlying number fields all behave in a uniform and highly structured way. The results of the project are expected to support future developments interrelating several disciplines surrounding number theory. The project includes collaborations with three undergraduates, designed specifically to better prepare them for scientific careers. These collaborations will serve as models for future involvement of students in high-level mathematics research.Hurwitz number fields arise as fields of definition of maps for Riemann surfaces to the Riemann sphere. The main focus of the research project will be on the explicit description of ramification in an arbitrary Hurwitz number field. From the geometric data defining the field, one can read off a finite set of primes outside of which ramification is known to be tame. The project will pursue exact formulas for ramification at these tame primes and sharp upper bounds for the wild ramification at the remaining primes. Besides studying ramification, the investigator will aim to establish a geometric analog of a conjecture on the unboundedness of degrees for a given set of ramifying primes. In a different direction, where the set of ramifying primes is allowed to grow, the investigator aims to demonstrate further uniformity of behavior by finding multi-indexed sequences of multivariate polynomials that define infinitely many Hurwitz number fields at once.